Quantum many-body scars (QMBS) are exceptional energy eigenstates of quantum many-body systems associated with violations of thermalization for special nonequilibrium initial states. Their various systematic constructions require fine-tuning of local Hamiltonian parameters. In this work, we demonstrate that long-range interacting quantum spin systems generically host robust QMBS. We analyze spectral properties upon raising the power-law decay exponent α of spin-spin interactions from the solvable permutationally symmetric limit α = 0 . First, we numerically establish that, despite the fact that spectral signatures of chaos appear for infinitesimal α , the towers of α = 0 energy eigenstates with large collective spin are smoothly deformed as α is increased and exhibit characteristic QMBS features. To elucidate the nature and fate of these states in larger systems, we introduce an analytical approach based on mapping the spin Hamiltonian onto a relativistic quantum rotor nonlinearly coupled to an extensive set of bosonic modes. We analytically solve for the eigenstates of this interacting impurity model by means of a novel polaron-type canonical transformation and show their self-consistent localization in large-spin sectors of the original Hamiltonian for 0 < α < d (with d = spatial dimension of the lattice ). Our theory unveils the stability mechanism of such QMBS for an arbitrary system size and predicts instances of its breakdown, e.g., near dynamical critical points or in the presence of semiclassical chaos, which we verify numerically in long-range quantum Ising chains. As a by-product, we find a predictive criterion for the presence or absence of heating under periodic driving for 0 < α < d , beyond existing Floquet-prethermalization theorems.
Quantum many-body scars (QMBS) are exceptional energy eigenstates of quantum many-body systems associated with violations of thermalization for special nonequilibrium initial states. Their various systematic constructions require fine-tuning of local Hamiltonian parameters. In this work, we demonstrate that long-range interacting quantum spin systems generically host robust QMBS. We analyze spectral properties upon raising the power-law decay exponent α of spin-spin interactions from the solvable permutationally symmetric limit α = 0 . First, we numerically establish that, despite the fact that spectral signatures of chaos appear for infinitesimal α , the towers of α = 0 energy eigenstates with large collective spin are smoothly deformed as α is increased and exhibit characteristic QMBS features. To elucidate the nature and fate of these states in larger systems, we introduce an analytical approach based on mapping the spin Hamiltonian onto a relativistic quantum rotor nonlinearly coupled to an extensive set of bosonic modes. We analytically solve for the eigenstates of this interacting impurity model by means of a novel polaron-type canonical transformation and show their self-consistent localization in large-spin sectors of the original Hamiltonian for 0 < α < d (with d = spatial dimension of the lattice ). Our theory unveils the stability mechanism of such QMBS for an arbitrary system size and predicts instances of its breakdown, e.g., near dynamical critical points or in the presence of semiclassical chaos, which we verify numerically in long-range quantum Ising chains. As a by-product, we find a predictive criterion for the presence or absence of heating under periodic driving for 0 < α < d , beyond existing Floquet-prethermalization theorems.
Quantum many-body scars are atypical energy eigenstates of chaotic quantum many-body systems that prevent certain special non-equilibrium initial conditions from thermalizing. We point out that quantum many-body scars exist for any nearest-neighbor spin-S XYZ quantum spin chain, and arise in the form of an infinite family of highly excited yet nonentangled product-state eigenstates, which define periodic textures in spin space. This set of scars, discovered originally by Granovskii and Zhedanov in 1985, encompasses both the experimentally relevant 'spin helices' for XXZ chains and more complicated helix-like states constructed from Jacobi elliptic functions for generic XYZ chains. An appealing feature of Granovskii-Zhedanov scars is that they are well-defined in the semiclassical limit S→∞, which allows for a systematic and analytical treatment of their dynamical instability to perturbations of the Hamiltonian. Using time-dependent spin-wave theory, we predict that upon perturbing along certain directions in Hamiltonian space, Granovskii-Zhedanov scars exhibit a dramatic asymmetry in their decay: depending on the sign of the perturbation, the decrease of their contrast is either slow and linear, or fast and exponential in time. This asymmetry can be traced to the absence (presence) of imaginarity in the spectrum of the Bogoliubov Hamiltonian governing quantum fluctuations about the scar, which corresponds to the absence (presence) of a non-zero Lyapunov exponent for the limiting classical trajectory. Numerical simulations using matrix product states (MPS) and infinite time-evolving block decimation (iTEBD) confirm that our prediction remains valid even far from the semiclassical limit. Our findings challenge existing theories of how quantum-many body scars relax.
This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.
Quantum many-body scars are special eigenstates that violate the eigenstate thermalization hypothesis while residing at finite energy density along with thermalizing eigenstates. The spin-1 XY model is known to host a family of such exceptional states originating from long-lived quasiparticle excitations that exhibit anomalously low entanglement entropy and long-time periodic revivals, resulting in weak ergodicity breaking. We study the stability of these scarred states against typical U(1) symmetry preserving perturbation in the XY chain. While perturbation theory can describe the deformed scar states at small system sizes, finite-size scaling of the perturbation matrix elements indicate that the scars ultimately thermalize in larger chains. Nonetheless, we demonstrate that the long-range order associated with the scars decays under the perturbation, and we estimate the relaxation timescale of oscillatory dynamics in certain local observables to be of order λ−2, where λ is the perturbation strength.
To probe the limits of dynamic thermalization, we numerically investigate the stability of exceptional periodic classical trajectories in chaotic many-spin systems and explore a possible connection between these trajectories and exceptional nonthermal quantum eigenstates known as "quantum many-body scars." The systems considered are chaotic spin chains with short-range interactions, both classical and quantum. On the classical side, the chosen periodic trajectories are such that all spins instantaneously point in the same direction, which evolves as a function of time. We find that the largest Lyapunov exponents characterizing the stability of these trajectories have surprisingly strong and nontrivial dependencies on the interaction constants and chain lengths. In particular, we identify rather long spin chains, where the above periodic trajectories are Lyapunov stable on many-body energy shells overwhelmingly dominated by chaotic motion. We show that the above phenomenology can be quantitatively described by connecting Lyapunov instabilities of translationally invariant periodic trajectories to irreducible representations of the translational symmetry group with well-defined wave vectors. We also find that instabilities around periodic trajectories in modestly large spin chains develop into a transient nearly quasiperiodic nonergodic regime. In some cases, the lifetime of this regime is extremely long, which we interpret as a manifestation of Arnold diffusion in the vicinity of integrable dynamics. On the quantum side, we numerically investigate the dynamics of quantum states starting with all spins initially pointing in the same direction: These are the quantum counterparts of the initial conditions for the above periodic classical trajectories. Our investigation reveals the existence of quantum many-body scars for numerically accessible finite chains of spins-3/2 and higher. No evidence of quantum scars was observed for spin-1/2 chains, while spin-1 chains were found to be transitional in this respect. The dynamic thermalization process dominated by quantum scars is shown to exhibit a slowdown in comparison with generic thermalization at the same energy. Finally, we identify quantum signatures of the proximity to a classical separatrix of the periodic motion.
We consider a quantum lattice spin model featuring exact quasiparticle towers of eigenstates with low entanglement at finite size, known as quantum many-body scars (QMBS). We show that the states in the neighboring part of the energy spectrum can be superposed to construct entire families of low-entanglement states whose energy variance decreases asymptotically to zero as the lattice size is increased. As a consequence, they have a relaxation time that diverges in the thermodynamic limit, and therefore exhibit the typical behavior of exact QMBS, although they are not exact eigenstates of the Hamiltonian for any finite size. We refer to such states as asymptotic QMBS. These states are orthogonal to any exact QMBS at any finite size, and their existence shows that the presence of an exact QMBS leaves important signatures of nonthermalness in the rest of the spectrum; therefore, QMBS-like phenomena can hide in what is typically considered the thermal part of the spectrum. We support our study using numerical simulations in the spin-1 XY model, a paradigmatic model for QMBS, and we conclude by presenting a weak perturbation of the model that destroys the exact QMBS while keeping the asymptotic QMBS.
The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite-energy density, leads to an absence of ergodicity and long-time coherent dynamics starting from initial states that have a high overlap with scars as experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and weak ergodicity breaking regimes. The former regime shows rapid thermalization of correlation functions and absence of scars in the Floquet spectrum of the system. The latter regime, in contrast, has scars; they lead to long-time coherent dynamics of correlation functions. We provide an analytical explanation of the existence of these regimes by going beyond a high-frequency Magnus expansion and using a novel perturbative approach valid at large drive amplitudes to derive a Floquet Hamiltonian which qualitatively explains the behavior of the driven system at arbitrary frequencies. We also discuss experiments involving finite Rydberg chains which can validate our theory. Our results demonstrate the possibility of drive-frequency-induced tuning between ergodic and weak ergodicity breaking dynamics in a disorder-free quantum many-body system.